Show that i begin bmatrix 5 -1116 - 7lendbmatrix bmatrix 2 - 1113 - 411 e n d b m a t r i x bmatrix 2 - 1113 - 4 1lendbmatrix bmatrix 8-1 16 - 7 1 e n d b m a t r i x ii bmatrix 1 - 2 - 3 0 - 1 - 0 11 - 1 - 0lendbmatrix bmatrix -1 - 1 - 0 10 -1 - 1 12 - 3 - 4 lendbmatrix -neq bmatrix -1 - 180110 -1 - 1 128384lendbmatrix -eginbmatrix 182 - 3 10 - 1 - 0 1

[ 5 amp; 1; 6 amp; 7 ][ 2 amp; 1; 3 amp; 4; ]= [ 10 3 amp; 5 4; 12+21 amp; 6+28; ]=[ 7 amp; 1; 33 ...

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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

Show that i b...

Question

Show that $(i) [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] \neq [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}]$

$(i i) ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ \neq ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$

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Solution

$[\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] = [\begin{matrix} 10 - 3 & 5 - 4 12 + 21 & 6 + 28 \end{matrix}] = [\begin{matrix} 7 & 1 33 & 34 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] = [\begin{matrix} 10 + 6 & - 2 + 7 15 + 24 & - 3 + 28 \end{matrix}] = [\begin{matrix} 16 & 5 39 & 25 \end{matrix}] H e n c e, [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] \neq [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}]$

Here, $⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 + 0 + 6 & 1 - 2 + 9 & 0 + 2 + 12 0 + 0 + 0 & 0 + (- 1) + 0 & 0 + 1 + 0 - 1 + 0 + 0 & 1 - 1 + 0 & 0 + 1 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 5 & 8 & 14 0 & - 1 & 1 - 1 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ a n d ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} - 1 + 0 + 0 & - 2 + 1 + 0 & - 3 + 0 + 0 0 + 0 + 1 & 0 - 1 + 1 & 0 + 0 + 0 2 + 0 + 4 & 4 + 3 + 4 & 6 + 0 + 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 1 & - 3 1 & 0 & 0 6 & 11 & 6 \end{matrix} ⎤ ⎥ ⎦$
Hence, the required results are verified.

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Show that $(i i) ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ \neq ⎡ ⎢ ⎣ \begin{matrix} - 1 & 1 & 0 0 & - 1 & 1 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$

Wherever possible, write each of the following as a single matrix

$(i) [\begin{matrix} 1 & 2 3 & 4 \end{matrix}] + [\begin{matrix} - 1 & - 2 1 & - 7 \end{matrix}]$
$(i i) [\begin{matrix} 2 & 3 & 4 4 & 6 & 7 \end{matrix}] - [\begin{matrix} 0 & 2 & 3 6 & - 1 & 0 \end{matrix}]$
$(i i i) [\begin{matrix} 0 & 1 & 2 4 & 6 & 7 \end{matrix}] + [\begin{matrix} 3 & 4 6 & 8 \end{matrix}]$

Let $A = [\begin{matrix} 2 & 1 0 & - 2 \end{matrix}], B = [\begin{matrix} 4 & 1 - 3 & - 2 \end{matrix}]$ and $C = [\begin{matrix} - 3 & 2 - 1 & 4 \end{matrix}]$ Find: $A^{2} + A C - 5 B$

If $A = [\begin{matrix} 3 & - 4 1 & - 1 \end{matrix}]$ , then prove that $A^{n} = [\begin{matrix} 1 + 2 n & - 4 n n & 1 - 2 n \end{matrix}]$ , where n is any positive integer.

Find X and Y if
$(i) X + Y = [\begin{matrix} 7 & 0 2 & 5 \end{matrix}]$ and $X - Y = [\begin{matrix} 3 & 0 - 1 & 5 \end{matrix}]$

$(i i) 2 X + 3 Y = [\begin{matrix} 2 & 3 4 & 0 \end{matrix}] a n d 3 X + 2 Y = [\begin{matrix} 2 & - 2 - 1 & 5 \end{matrix}]$

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